A048635 Number of rational points of Klein curve over GF(2^n).

0, 14, 24, 14, 0, 38, 168, 350, 528, 854, 1848, 4238, 8736, 16646, 31944, 64190, 131376, 265142, 526680, 1044974, 2088768, 4193126, 8404200, 16795166, 33541200, 67059734, 134195064, 268511054, 536991840, 1073711558, 2147211528

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = 14*x^2 + 24*x^3 + 14*x^4 + 38*x^6 + 168*x^7 + 350*x^8 + 528*x^9 + ...
		

References

• N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. See p. 77 eq. (3.13), (3.14).

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-4).

Crossrefs

Cf. A002249.

Programs

Mathematica

LinearRecurrence[{4,-7,8,-4},{0,14,24,14},40] (* Harvey P. Dale, May 09 2017 *)

PARI

{a(n) = if( n<1, 0, 2^n + 1 - 3 * polsym(x^2 - x + 2, n)[n+1])}; /* Michael Somos, Nov 09 2014 */

Formula

a(n) = 2^n + 1 - 3*(a^n + b^n), where a, b are roots of X^2 - X + 2 = 0.

From Colin Barker, Aug 01 2013: (Start)

a(n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 4*a(n-4).

G.f.: 2*x^2*(8*x^2-16*x+7) / ((x-1)*(2*x-1)*(2*x^2-x+1)). (End)